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34                        ANALYTICAL MECHANICS
theorem is illustrated in Fig. 24, where the direction of the translation is perpendicular to the axis of rotation.
39. Theorem IV.—The most general displacement of a rigid body can be obtained by a displacement similar to that of a screw in its nut, that is, by a rotation about an «.n,s and a translation along it.
This theorem states that the axis of rotation of the hist theorem can be so chosen that the translation is along the axis of rotation. In theorem III let PP', Fig. 25, be the put h of any point of the body described during the translation and BB be the line about which the body is rotated. Draw CC through P parallel to BB and drop the perpendicular P'P" upon CC. The displacement may be accomplished now in the following three stages. First: translate the \
body along the line CC until      !C              ,      „           B
the point  which was  at  P                     -M(" "'"'
arrives at P". Second: translate the body along />"/'' until the point arrives at P'. Third: rotate the body about Jilt until it comes to the desired position. But by theorem II the last two operations can be accomplished by a single* rotation about CC. Therefore the desired displacement can ho < >h-tainedby a translation along and a rotation about the lino CC.
Evidently the last theorem holds for infinitesimal displacements as well as for finite displacements; therefore however complicated the motion of a rigid body it ean ho reproduced by a succession of infinitesimal screw-duplacc-ments, each displacement taking the body from one position which it has occupied during the motion to another position infinitely near it. Thus at every instant of its motion the rigid body is displaced like a screw in its nut. In general